Longtime dynamics of Boussinesq type equations with fractional damping

Yang, Zuodong; Ding, Peng-Xu

doi:10.1016/j.na.2017.05.015

Cited by 11 publications

(11 citation statements)

References 29 publications

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“…Repeating the same arguments as in [32] shows that E(ξ u (0)) = E * , i.e., (13) holds for each t ≥ s ≥ 0. (iii) (Strong continuity and stability) By using energy identity (13) and repeating the same proof as in [32], one easily gets…”

Section: Zhijian Yang Pengyan Ding and Xiaobin Liumentioning

confidence: 91%

“…And by virtue of additional regularity of the weak solutions of Eq. (4), the interpolation theorem and the technique used in [32], we easily recover the topology of the phase space X and show that A α exp is just an exponential attractor of the dynamical system (S(t), X).…”

Section: Zhijian Yang Pengyan Ding and Xiaobin Liumentioning

confidence: 95%

“…(1), i.e., it makes them be of additionally partial smoothness when t > 0. Recently, Yang et al [31,32] showed that when α ∈ [1, 2) (structural damping case), Eq. (1) is like parabolic, the related solution semigroup has a global attractor and a "partially strong" exponential attractor in natural energy space when the growth exponent p of f (u) is up to the supercritical range: 1 ≤ p <p α ≡ N +2(2α −1) (N −2(2α−1)) + , where a + = max{a, 0}.…”

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confidence: 99%

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Attractors and their stability on Boussinesq type equations with gentle dissipation

Yang¹,

Ding²,

Liu³

2019

Communications on Pure &Amp; Applied Analysis

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The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation: utt + ∆ 2 u + (−∆) α ut − ∆f (u) = g(x), with α ∈ (0, 1). For general bounded domain Ω ⊂ R N (N ≥ 1), we show that there exists a critical exponent pα ≡ N +2(2α−1) (N −2) + depending on the dissipative index α such that when the growth p of the nonlinearity f (u) is up to the range: 1 ≤ p < pα, (i) the weak solutions of the equations are of additionally global smoothness when t > 0; (ii) the related dynamical system possesses a global attractor Aα and an exponential attractor A α exp in natural energy space for each α ∈ (0, 1), respectively; (iii) the family of global attractors {Aα} is upper semicontinuous at each point α 0 ∈ (0, 1], i.e., for any neighborhood U of Aα 0 , Aα ⊂ U when |α − α 0 | 1. These results extend those for structural damping case: α ∈ [1, 2) in [31, 32].

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Section: Zhijian Yang Pengyan Ding and Xiaobin Liumentioning

confidence: 91%

Section: Zhijian Yang Pengyan Ding and Xiaobin Liumentioning

confidence: 95%

mentioning

confidence: 99%

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Attractors and their stability on Boussinesq type equations with gentle dissipation

Yang¹,

Ding²,

Liu³

2019

Communications on Pure &Amp; Applied Analysis

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“…For example, it can be used to describe some physical, particularly mechanical, phenomena such as the paths of motion of celestial bodies and the brachistochrone problem [5,7]. Therefore, the dynamics of solution of the equation is paid close attention to by many researchers [9,17,[19][20][21][22][23]. For example, Sugiyama obtained a singularity formation for the 1D compressible Euler equations with variable damping coefficient [17].…”

Section: Introductionmentioning

confidence: 99%

“…For example, Sugiyama obtained a singularity formation for the 1D compressible Euler equations with variable damping coefficient [17]. Yang and Ding investigated the longtime dynamics of Boussinesq type equations with fractional damping [23]. The phenomenon of oscillations is observed in physics, economic, etc.…”

Section: Introductionmentioning

confidence: 99%

Philos-type oscillation criteria for second-order linear impulsive differential equation with damping

et al. 2019

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In this paper, the problem of oscillation for a second-order linear impulsive differential equation with damping is investigated. This equation can be more accurately used to model the states of many evolutionary processes, which are often subject to short-term perturbations and experience abrupt changes at certain moments of time. By employing a generalized Riccati transformation technique, we derive several oscillation criteria which are either new or improve several recent results in the literature. In addition, we provide an example to illustrate the effect of impulses on the oscillation of the equation.

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Asymptotic behavior of the Boussinesq equation with nonlocal weak damping and arbitrary growth nonlinear function

Ma,

Mo,

Wang

et al. 2024

Math Methods in App Sciences

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In this paper, we consider the asymptotic behavior of the Boussinesq equation with nonlocal weak damping when the nonlinear function is arbitrary polynomial growth. We firstly prove the well‐posedness of solution by means of the monotone operator theory. At the same time, we obtain the dissipative property of the dynamical system associated with the problem in the space and , respectively. After that, the asymptotic smoothness of the dynamical system and the further quasi‐stability are demonstrated by the energy reconstruction method. Finally, we show not only the existence of the finite global attractor but also the existence of the generalized exponential attractor.

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Longtime dynamics of Boussinesq type equations with fractional damping

Cited by 11 publications

References 29 publications

Attractors and their stability on Boussinesq type equations with gentle dissipation

Attractors and their stability on Boussinesq type equations with gentle dissipation

Philos-type oscillation criteria for second-order linear impulsive differential equation with damping

Asymptotic behavior of the Boussinesq equation with nonlocal weak damping and arbitrary growth nonlinear function

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